orthogonal vectors of vector $(a,b)$ which is of same length as that of vector $(a,b)$

Question

We have a vector $(a',b')$ which is orthogonal to vector $(a,b)$ and has same length as that of vector $(a,b)$ and lies counterclockwise of vector $(a,b)$.If vector $(a,b)$ is represented by $1i+ 1j$ .Then what should be vector $(a',b')$? How do we solve for vector $(a',b')$?

Answer 1

M and M' is orthogonal since the angle between two vectors is 90 degrees. Just rotate your point 90 degrees in clockwise direction

Answer 2

Might not be what you looking for, but here's an alternative approach exploiting the ease of rotating in the complex plane.

Identify $(a,b)$ with the complex number $z=a+bi$. The vector orthogonal to $(a,b)$ in counterclockwise direction, is the vector you get by rotating over 90° in the same direction. In the complex plane, this rotation corresponds to the simple multiplication with $i$, so the corresponding complex number is: $$z' = iz = i(a+bi) = bi^2+ia = -b+ai$$ and this corresponds to the vector $(a',b') = (-b,a)$.